i will try to make my question clear. Whe know the existence of $\phi=1.61803398874989484_\ldots$ or $\pi=3.14159265359_\ldots$ and we know that the decimals numbers goes to infinity.

But, let's say i write a randon number like : $$0.0031881208316410990203632024503368214675523306840544$$ Does it exists? Wolfram Alpha engine says it is : $$\frac{20751887915564}{6509128421234234}$$ But this engine cant handle well some others numbers like : $$0.2009999978498162165563223555516789561561$$

The question is : if i write any number with , lets say a billion randon decimals, this number will 'exist'?

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    $\begingroup$ Yes it will exist. It will also be rational. $\endgroup$
    – Mathmo123
    Jul 15, 2014 at 15:54
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    $\begingroup$ What is your definition of "existence"? Without this information, the question does not make any sense. $\endgroup$
    – user122283
    Jul 15, 2014 at 15:56
  • $\begingroup$ You seem to be misunderstanding what "existence" is mathematically. It is not a physical property, and you don't have to write it down to show that it exists (in classical mathematics anyway). So the question is a bit awkward, mathematically. $\endgroup$
    – Asaf Karagila
    Jul 15, 2014 at 16:19
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    $\begingroup$ You seem to be concerned with the possibility that you might write a number that doesn't exist. But, if it doesn't exist, then what would it even mean to write it? What's the "it" here? $\endgroup$ Jul 15, 2014 at 16:19
  • $\begingroup$ Sorry!!! my point was : for example, is there a infinity amout of numbers between 0,002 and 0,001 ? For exemple $$0.001656822229448781$$ (and a infinite amount of digits after the decimal) or $$0.0010008466697$$ (and a infinity amount of digits after the decimal) $$.$$ I Wanna know if "between" x and y exisits a infinity amout of numbers with infinity digits after the decimal $\endgroup$
    – mateuszica
    Jul 15, 2014 at 16:54

2 Answers 2


My understanding of your question is: does any number that can be written with a finite amount of digits, can be written in the form $\frac{a}{b}$ with $a \in \mathbb{Z}, b \in \mathbb{N}, b \neq 0$. The answer is yes!

Suppose $x$ can be written with a finite amount of digits (say $n$ digits). Then $$x = \frac{x\cdot 10^n}{10^n} = \frac{a}{b},$$ and $a\in \mathbb{Z},b \in \mathbb{N}$. Of course this fraction may not be irreducible but it is still a "rational representation" of your number.


If you write a number with exactly 1 billion digits after the decimal, it exists because it can be expressed as the rational number (number without the decimal)/(10^1-billion).

Similarly any terminating decimal can be expressed as a rational by removing the decimal point and dividing by 10^(# of decimal places).

Even for an infinite decimal we know it exists. This is because the (standard) definition of a real number is an convergent sequence of rational numbers. The sequence we get by continuously adding another decimal place is convergent and so the number exists.


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